Astronomy & Astrophysics manuscript no. paper c ESO 2020 June 25, 2020

Velocity limits in the thermonuclear ejection scenario for hypervelocity stars and the origin of US 708 P. Neunteufel1,2

1Max Planck Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching bei München 2University of Leicester, University Road, LE1 7RH Leicester, Leicestershire e-mail: [email protected] Received (month) (day), (); accepted (month) (day), (year)

ABSTRACT

Context. Hypervelocity stars (HVS) are a class of stars moving at high enough velocities to be gravitationally unbound from the . In recent , ejection from a close binary system in which one of the components undergoes a thermonuclear supernova (SN) has emerged as a promising candidate production mechanism for the least massive specimens of this class. The explosion mechanisms leading to thermonuclear supernovae, which include the important Type Ia, and related subtypes, remain unclear. Aims. This study presents a thorough theoretical analysis of candidate progenitor systems of thermonuclear SNe in the single de- generate helium donor scenario in the relevant parameter space leading to the ejection of HVS. The primary goal is investigation of the, previously unclear, characteristics of the velocity spectra of the ejected component, including possible maxima and minima, con- straints arising from stellar evolution and initial masses. Further, the question of whether knowledge of the ejection velocity spectra may aid in reconstruction of the terminal state of the supernova progenitor, is addressed, with a focus on the observed object US 708. Methods. Presented are the results of 390 binary model sequences computed with the Modules for Experiments in Stellar Astrophysics (MESA) framework, investigating the evolution of supernova progenitors composed of a helium-rich hot and a accreting . Assumption of a specific explosion mechanism is avoided as far as possible. The detailed evolution of the donor star as well as radiation and mass transfer driven orbital evolution are fully taken into account. Results are then correlated with an idealized kinematic analysis of the observed object US 708. Results. It is seen that the ejection velocity spectra reach a maximum in the range 0.19 M < MHVS < 0.25 M . Depending on the local Galactic potential, all donors below 0.4 M are expected to become HVS. The single degenerate helium donor channel is able to account for runaway velocities up to ∼ 1150 km s−1 with a Chandrasekhar mass accretor, exceeding 1200 km s−1 if super- Chandrasekhar mass detonations are taken into account. It is found that the previously assumed mass of 0.3 M for US 708, combined with more recently obtained proper motions, favor a sub-Chandrasekhar mass explosion with a terminal WD mass between 1.1 M and 1.2 M , while a Chandrasekhar mass explosion requires a mass of > 0.34 M for US 708. This mechanism may be a source of isolated runaway extremely low mass white dwarfs. Conclusions. The presence of clear, terminal accretor mass dependent, but initial-condition independent, ejection velocity maxima, provides constraints on the terminal state of a supernova progenitor. Depending on the accuracy of , it is possible to discern certain types of explosion mechanisms from the inferred ejection velocities alone, with current proper motions allowing for a sub- Chandrasekhar mass SN as an origin of US 708. However, more robust reconstructions of the most likely SN progenitor state will require a greater number of observed objects than are currently available. Key words. supernovae: general – white dwarfs – (Stars:) binaries (including multiple): close – (Stars:) subdwarfs – (Stars:) white dwarfs

1. Introduction led to a targeted search, conducted by Brown et al. (2007), which yielded nine further objects in short order, growing to 23 The existence of stars moving at velocities high enough to be objects within the following decade (Brown 2015). unbound from the Galaxy, known as hypervelocity stars (HVS), Theoretical modeling of the ejection mechanism at this point was first proposed more than three decades ago by Hills (1988). in time focused heavily on the connection. Bromley arXiv:2006.11427v2 [astro-ph.SR] 24 Jun 2020 While this initial prediction was followed up by a number of et al. (2006) predicted the likely ejection velocity spectrum of theoretical studies (Hills 1991, 1992), primarily based on the as- HVS produced by the encounter of a binary star with a super sumption that these object would result from interaction of a star massive black hole (SMBH) at the Galactic center (Sag A∗). or binary star with a (Yu & Tremaine 2003), massive black hole O’Leary & Loeb (2008) pointed out that an encounter with a (MBH), observational evidence was not forthcoming. stellar mass black hole in orbit around Sag A∗ could also act as a This changed with the discovery of SDSS source of HVS with Sesana et al. (2009) coming to the same con- J090745.0+024507 by Brown et al. (2005). Found in the clusion regarding intermediate mass black holes (IMBH). The data set, this object, also referred to latter mechanism has been suggested as a source of certain B- as HVS1, is likely a B-type star with a mass of about 3 M at a type HVS (Irrgang et al. 2019). Very recently, discovery of an distance of 111 kpc from the Galactic center and a Galactic rest A-type star of 2.35 M , moving at a Galactic rest frame velocity frame velocity of 696 km s−1 (Brown et al. 2007). This object of ∼ 1700 km s−1, and which could be traced with high accuracy

Article number, page 1 of 17 A&A proofs: manuscript no. paper back to the Galactic center (Koposov et al. 2020), has greatly available for close white dwarf binary systems, including initial bolstered the viability of the SMBH-encounter scenario. For a mass and mass ratio, as well as initial orbital separation, pro- review of the state of the art on this ejection mechanism, the posed to allow for the occurrence of thermonuclear SNe. As the reader is directed to Brown (2015). precise explosion mechanism of these events is currently still Alternative ejection mechanisms to black hole interaction unresolved, the methodology aims to remain agnostic to it and, started being considered in 2009 when Abadi et al. (2009) pro- therefore, to the precise state of the accretor at the time of cre- posed, based on the spatial distribution of known HVS, an ori- ation of the HVS. This paper presents the results of 390 detailed gin in multiple body interaction during the passing of a satellite binary evolution models sequences, commenting on the ejection dwarf galaxy through the . The proposal by Justham velocity spectra of a variety of proposed explosion mechanisms et al. (2009) that some HVS could be former members of close of thermonuclear SNe, the pre-explosion evolution of the system binary system with a white dwarf (WD) companion undergo- and the donor star and the viability of using HVS as probes of ing a supernova (SN) explosion provided a conceptual bridge thermonuclear SN explosion mechanisms. to ejection mechanisms earlier proposed for runaway stars (RS) It should be emphasized that the focus of this study is HVS gravitationally bound to the Galaxy (Blaauw 1961; Hoogerw- ejection by thermonuclear SNe in general, of which Type Ia SNe erf et al. 2001), earlier considered to be incapable of providing are considered a subtype, not Type Ia SNe exclusively, whose the necessary ejection velocities. This idea was later followed relatively similar peak luminosity hints at a likewise relatively up theoretically for both B and G/K-dwarf type stars in core col- similar terminal WD mass at the point of explosion (Phillips lapse SNe (Tauris 2015) and hot subdwarfs in the thermonuclear 1993). However, in the absence of a consensus on the spectral scenario (Neunteufel et al. 2016, 2019). classification of hypothetical progenitor SNe of observed HVS, Observational evidence for the viability of the supernova it is reasonable to assume that WD masses at the point of deto- ejection scenario came when Geier et al. (2015) pointed out nation in these events may be dissimilar to those responsible for that the, previously known (Hirsch et al. 2005), helium sdO, of Type Ia SNe. This paper can make no statement on the obser- N-type, indicating unusually fast rotation, US 708 (HVS2) was vational properties of the SN event, only on the velocity of the moving with a greater velocity than originally reported and could expected runaway for a certain assumed WD mass. not be traced back to the Galactic center. This report is organized as follows: Sec. 2 presents a brief re- More recently, accompanying observational successes (e.g. view of the literature concerning explosion mechanisms of ther- Raddi et al. 2019), theoretical interest has focused on the evo- monuclear SNe. In Sec. 3, a number of analytical considera- lution of an SN-ejected HVS in the phase after the supernova tions relevant for the investigation of close binary systems are explosion (Zhang et al. 2019; Bauer et al. 2019), both under the presented. Sec. 4 comments on the numerical tools used in this assumption of the HVS being the former donor star and or a par- project and justifies the choice of initial model parameters. Sec. 5 tially burnt remnant of the former accretor. presents the study’s findings, commenting on the bulk properties With the advent of the latest generation of large scale astro- of the sample and observational properties of certain individual metric surveys (Gaia Collaboration et al. 2018), a number of new cases. Sec. 6 presents a simple application to the observed object objects (Shen et al. 2018), thought to originate from a thermonu- US 708, of the predictions in the preceding sections. In Sec. 7, clear supernova occurring in a double WD system, as well as a results are discussed, with a brief summary and conclusions in number of additional candidates (de la Fuente Marcos & de la Sec. 8. A brief investigation of the effects of variations in the Fuente Marcos 2019), have been discovered. In the same vein, initial orbital separation is shown in Appendix A. Sec. 6 makes data obtained by the Gaia instrument has confirmed the origin extensive use of calculations of the motion of a hypervelocity of HVS3 (HE 0437-5439) in the Large Magellanic cloud (Edel- star in the Galactic potential. As this is slightly outside the focus mann et al. 2005; Irrgang et al. 2018; Erkal et al. 2019). Con- of this study, the potential and numerical tool employed here are sidering the capabilities of upcoming instruments (e.g. 4MOST: briefly discussed in Appendix B. de Jong et al. 2016), further discoveries in this field may be ex- pected. The supernova ejection scenario for HVS can be considered 2. Review of applicable explosion mechanisms of particular attractiveness, since, as will be shown, the ejection While it is largely accepted that thermonuclear SNe result from velocity spectrum for these objects is closely related to the pro- the thermonuclear detonation of a WD receiving material from genitor binary’s orbital parameters and mass distribution at the a binary companion, the nature of the progenitor system is cur- point of HVS ejection. Knowledge of these parameters, which rently not well understood. Generally, hypotheses regarding the contain information of the state of the exploding companion, companion fall into two distinct categories: Double degenerate can then be used to infer the explosion mechanism of the super- (DD), where the companion in another WD, and single degen- nova, which, in the case of thermonuclear events, is still not con- erate (SD), where the companion is a non-degenerate star (e.g. a clusively resolved (see, e.g., review by Hillebrandt & Niemeyer main sequence star)1. However, apart from the reasonably well 2000). established identity of the exploding object as a WD, uncertain- Studies of the pre-explosion evolution of close binary sys- ties persist as to its state, especially its mass, at the point of det- tems have been performed in the past (e.g. Ergma & Fedorova onation (terminal mass) as well as the outcome of the explosion. 1990; Yoon & Langer 2004a,b; Yungelson 2008; Wang et al. It is classically understood that in systems containing two WDs 2013; Neunteufel et al. 2016, 2019), with a focus not on run- of sufficient mass, Roche lobe overflow (RLOF) leads to one away velocities but on the ignition behavior of the accretor. How- or both companions being dynamically disrupted before merg- ever, while Han (2008) and Wang & Han (2009) did study the ing, with the merged object detonating (see e.g Webbink 1984). ejection velocity distribution of donor stars ejected from these systems subsequent to a supernova explosion using a population 1 Note that for our purposes the defining characteristic for categoriza- synthesis framework, their parameter space is limited to runaway tion into these groups consists solely in the interaction between either masses ≥ 0.6 M . The present study is meant to remedy this sit- two degenerate objects or one degenerate and one non-degenerate ob- uation, presenting a detailed examination of the parameter space ject (compare Soker 2013).

Article number, page 2 of 17 P. Neunteufel: Hypervelocity star ejection

While it is generally expected that this scenario will leave no in a detonation) models tend to produce insufficient amounts of bound remnant, certain violent merger scenarios have been pre- IGE in addition to featuring insufficiently high ejecta velocities. dicted to occur on short enough a timescale to allow for the ejec- tion of a bound object (Pakmor et al. 2013; Shen et al. 2018). A method to overcome this dichotomy was proposed in the Note that dynamical disruption of the mass donor can be avoided deflagration-detonation scenario (DDT), which presupposes ig- in double degenerate systems with relatively small (q ≤ 0.63) nition of core carbon burning in the subsonic regime, transition- mass ratios (Eggleton 2011). However, as we are concerned with ing to the supersonic regime during the time the burning front the production of non-degenerate HVS, the DD scenario will be takes to traverse the WD (Khokhlov et al. 1997). While this sce- disregarded. In the SD-scenario, the binary is not expected to nario generally produces isotope yields well in agreement with merge. Instead, the non-degenerate companion will donate ma- observation, major challenges persist in the self-consistent mod- terial to the WD (note that this mass transfer is, depending on eling of the subsonic-supersonic transition, as well as that of the the mass ratio of the system and the evolutionary state of the microphysics involved in the propagation of the burning front. donor, not necessarily stable). Depending on the prevalent ex- Further uncertainties persist in whether the deflagration, plosion mechanism, which, as of this time, is still heavily de- once initiated, develops into a detonation after an initial delay bated, a thermonuclear explosion is initiated on the WD by one (the delayed DDT scenario) - prompt transition into the detona- of the mechanisms discussed below as soon as the requisite ig- tion regime would be akin to the prompt scenario describe above nition conditions are reached. If the WD is completely disrupted - or after a number of "pulses" (the pulsational DDT scenario). by the explosion, the donor star is, under preservation of its or- These pulses would see one or more increases and subsequent bital angular momentum, flung away from the former location drops in nuclear burning, accompanied by expansion and con- of the binary with a velocity slightly greater than its terminal traction of the WD, with CO being converted into IMEs, before orbital velocity (Bauer et al. 2019). It should be noted that not the final ignition of a detonation which would then be visible as all hypotheses concerning the mechanism of thermonuclear SNe an SN. predict the complete disruption of the WD, instead leaving a par- tially burnt remnant (Vennes et al. 2017). A number of objects Also included in this category are certain outcomes of sys- fitting this scenario have recently been described (Raddi et al. tems resulting in a double detonation scenario (this mechanism 2019). This would imply that the runaway velocity will be lower is more relevant for the sub-Chandrasekhar category and is dis- than in the case of a complete disruption of the WD (with a pos- cussed in greater detail in that context. See Sec. 2.2) as they con- sibility of the partially burnt remnant becoming a HVS). While tinue to accrete to within 0.05 M of the Chandrasekhar mass doubtlessly important, closer study of this case is beyond the (Neunteufel et al. 2019), depending on the efficiency of angular scope of this paper and is left for future inquiry. As will be fur- momentum transport in the accreting WD. ther commented on in Sec. 3, obtainable velocities in the SD As mechanisms acting at this terminal mass can be con- supernova ejection mechanism are inversely correlated with the sidered "classical", observed objects moving with velocities in- physical radius of the donor star. This circumstance suggests that compatible with terminal masses in the range 1.35 M > Md,f > helium-rich donor stars in their core helium burning phase are 1.45 M will be most interesting. the principal candidates for the production of non-degenerate HVS in the supernova ejection scenario. This study therefore chiefly considers production mechanisms for thermonuclear SNe 2.2. Sub-Chandrasekhar mass mechanisms relying on accretion from non-degenerate He-rich donor stars in their core helium burning phase and assumes that the explosion of the WD will leave no appreciable bound remnant. If thermonuclear SNe can be ignited well below the Chan- drasekhar mass, then the initiation of nuclear burning will not be Keeping this in mind, as well as the importance of the ter- related to the WDs stability against gravitational collapse. The minal mass of the WD to the runaway velocity, we order our initial "spark" setting off the thermonuclear detonation of the given test cases into three broad categories: Chandrasekhar mass CO core will therefore not be lit at the center of the WD. One mechanisms, sub-Chandrasekhar mass mechanisms and super- mechanism capable of achieving this was first proposed in the Chandrasekhar mass mechanisms. The following discussion is 1980s by Nomoto (1980, 1982a). This mechanism, now widely conveniently summarized in Fig. 1 and related citations in Tab. 1, known as the double detonation (DDet) mechanism, posits that a showing likely terminal WD masses as proposed in literature at helium layer, accumulated from a helium-rich companion star the time of writing. of some description, on top of the WD’s CO core can act as a detonator for the thermonuclear disruption of the star. Under certain (currently relatively well but not completely understood) 2.1. Chandrasekhar mass mechanisms circumstances, ignition of nuclear burning in such a helium layer will lead to a detonation of the helium. The associated detona- Mechanisms falling into this category are distinguished by the tion shock will then trigger a secondary detonation of the carbon assumption that a thermonuclear SN is initiated when the ac- in the WD’s core. Terminal accretor masses in this scenario have creting WD reaches a terminal mass close to the Chandrasekhar been argued to be as low as 0.75 M (Livne & Arnett 1995). mass (Hoyle & Fowler 1960; Arnett 1969; Hansen & Wheeler 1969). While a number of successes in the spectral modeling of Another mechanism falling into this category, while very explosion mechanisms in this category could be achieved (see similar to the DDet mechanism is the prompt double detona- e.g. Nomoto et al. 1984; Hillebrandt & Niemeyer 2000), major tion (PDDet) or dynamically driven double degenerate double challenges persist in resolving the dominant one. Prompt deto- detonation (D6) mechanism. This mechanism relies on turbulent nation at the time terminal mass is attained is generally ruled out ignition of a thin < 0.05 M helium layer, accumulated dynami- (Arnett 1971) on the grounds of significant overproduction of cally from a companion (most likely a He-WD or hybrid HeCO iron group elements (IGE), while pure deflagration (i.e. subsonic WD), setting off a secondary detonation of the accretor’s CO flame propagation, as opposed to supersonic flame propagation core (Pakmor et al. 2013).

Article number, page 3 of 17 A&A proofs: manuscript no. paper

2.3. Super-Chandrasekhar mass mechanisms N1982 In the DD scenario, super-Chandrasekhar mass explosions are 7 WW1994 expected to occur simply if the total mass of the system, contain- YL2004a SRH2010 WK2011 ing two sub-Chandrasekhar mass WDs, is sufficiently high. In NYL2017 the SD scenario, however, theory requires super-Chandrasekhar 6 mass detonations to rely on rotation. The effects of rotation on the stability of WDs against gravitational collapse has been of ] 1 5 −

interest to the astrophysical community for some time. Specif- LA1995 YL2005 yr

ically, Yoon & Langer (2005) showed that fully differentially rotating WDs may avoid gravitational collapse at masses up to M · 4 2.2 M . Rigid rotation is unable to prevent gravitation collapse at 8 − masses higher than 1.5 M (Hachisu et al. 2012). In either case, PCL2004 PCL2004 [10

an accreting WD, starting at masses sufficiently below the Chan- ˙ 3 M GKO2005GKO2005 drasekhar mass to allow for the WD to be sufficiently spun up LCW2010 HSN2006 by angular momentum accretion to avoid collapse at the Chan- drasekhar mass can be expected to grow beyond it. After the 2 system detaches, the fast rotation of the WD can then be slowed FHR2007 through mechanisms like tidal interaction, the r-mode instabil- 1 ity (see e.g. Yoon & Langer 2004a) or, conceivably, magnetic braking (Mestel 1968). As the rotation of the WD slows (which is expected to occur on timescales of up to Gyrs), it becomes 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 progressively less stable against gravitational collapse, which is MWD,f [M ] expected to occur once the WD has lost a certain amount of angular momentum. This scenario, widely known as the spin- Fig. 1: Representation of the parameters of proposed explosion up/spin-down (su/sd) mechanism was first proposed some time mechanisms in the parameter space accessible to this study with ago (see e.g. Di Stefano et al. 2011), with significant uncertain- M˙ the mass transfer rate and MWD,f the proposed total mass of ties remaining related to explosion physics and the spin-down the accreting WD at the point of detonation. If both values are timescale. However, it is unlikely that a runaway hot subdwarf is provided by the source, each model is represented as a point. If produced if the spin-down timescale is very long. the source only provides a single value for MWD,f, the model is represented as a blue line. If a range for MWD,f is given, it is 3. Physical considerations represented as pairs of colored lines with arrows indicating the position of the corresponding second line. Labels indicate source The terminal orbital velocity of a donor star of a given mass Md material as defined in Table 1. and radius Rd in a binary system undergoing a thermonuclear supernova can be calculated by the widely-used approximation provided by Eggleton (1983)    (GWR, see e.g. Landau & Livshitz 1975). Of these, magnetic R  !2/3 !1/3  braking is thought to be important mainly in solar-type stars d,RL  MWD  Md  a = 0.6 + ln1 +  (1) and neglected in He rich stars like the ones under consideration 0.49  Md  MWD  here. As the systems under consideration here can be thought of with the assumption that the donor star exactly fills its Roche as circularized and tidally locked subsequent to the most recent lobe (i.e. the condition Rd = Rd,RL). The terminal orbital velocity common envelope (CE) phase ("most recent" referring to the CE then follow from the Keplerian equation phase immediately preceding the formation of the He donor). Tides, as well as the possibility of non-conservative mass trans- s 2 fer, are neglected. GMWD vej = (2) (MWD + Md)a The orbital evolution of these systems therefore dominated by mass transfer and GWR. While angular momentum loss due where MWD = Macc is the mass of the accretor, and G is the grav- to GWR acts to decrease orbital separation under all circum- itational constant. If the quantity Rd(Md) can be expressed ana- stances, the angular momentum transported during mass transfer lytically, the terminal orbital velocity follows immediately from may act to either increase or decrease orbital separation depend- Eqs. 1 and 2. However, as Rd(Md) realistically depends on the ing on the system’s mass ratio, with q = Md/Macc < 1 associated structure of the star as well as its current mass loss rate due to with increasing orbital separation and vice versa. In the systems RLOF, a consistent solution of both the stellar structure equa- under consideration, RLOF is induced either through an increase tions and the orbital evolution of the system, is required. of the donor star radius or decrease of the orbital separation due Unlike previous studies (e.g. Brooks et al. 2017; Neunteufel to GWR. Once the system evolves into a semidetached state, fur- et al. 2019), which included the detailed evolution of the accret- ther evolution of the orbit is driven by both GWR and angular ing companion, this study treats the accretor as a point mass. momentum transfer. This, consciously adopted, simplification allows maintenance of Eggleton (2011) gives for GWR, neglecting mass transfer a certain "agnosticism" towards the explosion mechanism and terminal state of the accretor. The orbital evolution in binary systems is generally in- fluenced by the effects of magnetic braking (Mestel 1968), a˙ 2 tides (Hut 1981), mass transfer and gravitational wave radiation = − (3) a τGR Article number, page 4 of 17 P. Neunteufel: Hypervelocity star ejection

Table 1: Sources of utilized test cases

Label Citation Method Additional Notes N1982 Nomoto (1982a,b) 1D SEC DDet mechanism proposed WW1994 Woosley & Weaver (1994) 1D SEC - LA1995 Livne & Arnett (1995) 2D HS - YL2004 Yoon & Langer (2004a) 1D SEC Included rotational instabilities PCL2004 Plewa et al. (2004) 2D HS - GKO2005 Gamezo et al. (2005) 3D HS Representative for the MCh-case YL2005 Yoon & Langer (2005) 1D SEC Super-MCh-case, differential rotation HSN2006 Howell et al. (2006) Observational Observed Super-MCh, inferred mass FHR2007 Fink et al. (2007) 3D HS - SRH2010 Sim et al. (2010) 1D HS - LCW2010 Liu et al. (2010) 1D SEC - WK2011 Woosley & Kasen (2011) 1D SEC+HS - NYL2017 Neunteufel et al. (2017) 1D SEC included rotation+magnetic torques

Notes. Sources for main test cases used in this study, including the methodology employed by each source: Stellar evolution calculation (SEC), hydrodynamical simulation (HS) or observations.

with a the semi-major axis and the gravitational merger where HP is the photospheric pressure scale height, R the stellar timescale radius as defined by the photosphere, and Rd,RL the Roche lobe 5 c5a4 radius with RRL calculated according to Eq. 1. τ = (4) GR 3 32 G (M1 + M2)M1 M2 This study is mostly concerned with the ejection velocity of donor stars ejected from systems with terminal accretor masses and for mass transfer, neglecting GWR in the range 1.1 M ≤ Mt,acc ≤ 1.5 M . In order to provide suffi- a˙ M˙ q2 − 1 cient coverage of the grid, initial accretor masses were chosen in 1 . = 2 (5) the range 0.5 M ≤ Mt,acc ≤ 1.2 M in steps of 0.05 M . Initial a M1 + M2 q donor star models range in mass between 0.4 M and 1.0 M Note that Eq. 5 implies that the evolution of the orbital separa- with solar metallicity. As per Eq. 1, large stellar radii will lead to tion is independent of the mass transfer rate if q = 1. Comparing lower ejection velocities. Limiting this study to likely production Eqs. 4 and 5 yields, with the assumption q < 1 and the demand mechanisms of hypervelocity stars originating in the Galactic thata ˙/a > 0 the condition disk, this condition excludes post-HeMS stars due to the require- 3 2 2 ment to exceed the Galactic during ejection. Cal- 32 G (M1 + M2)M M M˙ < M˙ = − 1 2 . (6) culations are therefore terminated if the donor stars reaches the 1 crit 5 4 5 c a M1 − M2 end of its core helium burning phase before the onset of RLOF. as derived by Tutukov & Yungelson (1979). Due to its inverse proportionality to the fourth power of a, Eq. 6 is usually fulfilled by default in most systems containing an ordinary star, whose 4.1. Initial models physical radius is on the order of one magnitude greater than that of a hydrogen depleted star of comparable mass. With grav- The initial states of the employed donor models are summa- itational merger timescales being comparable to the mass trans- rized in Fig. 2. These initial models are created by initializing a M˙ 1 fer timescales (τMT = ) in the systems under consideration hydrogen-depleted pre-MS model using the MESA-supplied op- M1+M2 here, the sign of the time derivative of the orbital separation will tion create_pre_main_sequence_model = .true., allow- be determined by Eq. 6. ing it to settle on the HeMS, relaxing it further for a thermal timescale (He-rich models like this tend to contract by 1-5% af- ter reaching the HeMS). The donor model is then placed in a bi- 4. Numerical methods and initial model parameters nary system with appropriate characteristics. Initial orbital sepa- The fundamental methodology of this study consists of the de- rations ainit(ξ) were chosen such that Eq. 1 satisfies Rd,RL = ξ ·Rd tailed calculation of the orbital evolution of close He-star+WD with ξ an arbitrary dimensionless parameter. Depending on the binaries. This is accomplished using the MESA framework (Pax- individual component masses within the given ranges, a system ton et al. 2011, 2013, 2015, 2018) in its release 11398. of the present configuration is not expected to interact during the MESA is publicly available and well established stellar and donor’s HeMS lifetime at all for orbital radii ainit(ξ > 1.01). The binary evolution code, capable of treating the evolution of sin- most viable choice is deemed to be ainit(ξ = 1.005) with a sec- gle stars as well as that of the orbital parameters of binary sys- ondary sample of ainit(ξ = 1.01) (see Appendix A). This increase tems. MESA offers a variety of prescriptions to calculate mass only results in a comparably small increase in the initial period loss due to RLOF. For systems of this type, the mass loss pre- of the system. However, as shown in Neunteufel et al. (2019), scription provided by Ritter (1988), implemented as MESA- donor stars of initial mass & 0.8 M tend to reach the end of core option mdot_scheme = ’Ritter’, is considered most appro- helium burning disproportionally quickly while those of lower priate. This scheme relies on implicitly solving mass . 0.5 M tend to evolve slowly enough to not have expe- rienced significant increase in mass or metallicity before reach- ! M˙ ing GWR-induced RLOF. While this approach somewhat limits − · Rd,RL R + HP ln = 0 (7) predictive power in intermediate masses, upper and lower limits M˙ 0 Article number, page 5 of 17 A&A proofs: manuscript no. paper should be adequately addressed. Initial periods resulting from 2.6 1.00 this prescription are shown in Fig. 2 (C). 0.95 2.4 0.90 0.85 (A) 2.2 . 4.2. Stopping conditions 0 80 2 0.75 ) 0.70 Computations are terminated either when the donor star’s core L 1.8 0.65 helium abundance decreases below Y = 10−3 or if the donor / core L 0.60 1.6 star’s mass drops below 0.18 M . The first condition is motivated 0.55 by the desire to focus on the production of hypervelocity hot sub- log( 1.4 0.50 dwarfs. Further, the coincident expansion of their envelopes will 1.2 result in increased mass transfer rates and, therefore, according 0.45 to Eq. 6, an increase in orbital separation and, consequently, de- 1 Labels indicate Minit in [M ] creased ejection velocities. While further investigation of these 0.8 0.40 hypothetical post-hot-subdwarf runaways is of some interest, it 2530354045505560 is beyond the scope of this paper. The second condition is war- ranted due to the limitations of the equation of state (EOS) as Teff [kK] Z · 2 Z Z · 0.032 currently implemented in MESA as donor star models in their proto-WD phase (i.e. M < 0.3M ) tend to cross into regions of 0.22 4.6 the parameter space with insufficient coverage (see Paxton et al. (B) 2019, Appendix A), resulting in numerical artifacts or unresolv- 0.2 4.5 able models. Specifically, these instabilities occur as sufficient 0.18 4.4 ] amounts of unburnt helium from the outer layers of the star are 3 0.16 4.3 −

removed, exposing the formerly burning and metal enriched core ] layers, which, at this point, will be cool and sparse enough lie 0.14 4.2 [R outside the coverage of the EOS. It is found that this problem is ) [g cm R ρ largely avoided by stopping the simulation at 0.18 M , as most 0.12 4.1 models with both more massive and significantly metal enriched 0.1 4 log( cores will have left the HeMS (and, consequently, been removed by the first stopping condition) at this point with remaining mod- 0.08 3.9 els possessing either less massive or sufficiently pristine cores. 0.06 3.8 0.4 0.5 0.6 0.7 0.8 0.9 1

5. Ejection velocity spectra M [M ] R ρ The principal aim of this study is to provide ejection velocity spectra for hypervelocity runaways produced in the He-star+WD 4.5 channel for thermonuclear SNe. An ejection velocity spectrum, 1.3 (C) 4 for the purposes of this paper, is composed of the expected ejec- 1.2 tion velocities of a runaway, depending on the terminal mass 1.1 3.5 of the runaway, for a single terminal WD mass. In order to re- ]

d] main unbiased towards the plethora of proposed explosion mech- 1 3 · [M 01 anisms, as discussed in Sec. 2, which would impose systematic 0.9 2.5 . constraints on the derived ejection velocity spectra, a range if [0

WD,init 0.8 terminal WD masses are taken into consideration. 2 init P M 0.7 1.5 5.1. Partial spectra 0.6 1 Contribution of individual binary model sequences to the ejec- 0.5 tion velocity spectra for any particular terminal WD mass 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 (MWD,f) are shown in Fig. 3. The entire spectrum for any par- ticular choice of MWD,f, as derived in this study is then com- Md,init [M ] posed of all binary system states with the same value of MWD,f. This means that, rather than being determined by the evolution Fig. 2: Relevant initial parameters of the utilized He-donor star of the orbital velocity of an individual binary model sequence, models. Panel (A) indicates the position of each model in the the ejection velocity spectrum is composed of the systems’ in HR-diagram, with labels corresponding to initial mass in units the vorb-Md parameter space where the mass of the accretor is of M . Colors indicate initial metallicity in units of the solar equal to the requested value of MWD,f across multiple model se- metallicity Z with solar metallicity used in this study. Panel (B) quences. In the partial spectra seen in Fig. 3, a clear correlation indicates the initial radius Rinit in units of the solar radius R between orbital velocities and the terminal mass of the accretor, and central density ρc with colors corresponding to metallicity independent of both initial masses is apparent. Individual tracks as in panel (A). Panel (C) shows initial periods, according to the exhibit a maximum of the orbital velocity, located in the range constraints discussed in Sec. 4.1 with respect to the initial masses 0.2 M < Md < 0.25 M . As the full ejection velocity spectra of both components. are essentially determined by the shapes of numerous individual tracks, this feature is also expected in the complete spectra.

Article number, page 6 of 17 P. Neunteufel: Hypervelocity star ejection

1200 0.14 M = 1.3M 1100 WD,f 0.13 M = 1.2M WD,f 0.12 1000 M = 1.1M WD,f Md = 0.5M 0.11 2 (WD) ] / 1 900 3 ] − 0.1 M = 0.55M

d,in 800 = [R (A) 0.09 MWD,in = 1.00M n d [km s 700 R 0.08 ej v 600 0.07 Rmin 500 0.06 Md = 0.3M M = 1.00M 0.05 400 WD,in 0.04 0.2 0.3 0.4 0.5 0.6 0.7 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Md [M ] ρ¯d/ρd,c 1200 Fig. 4: Radius of the donor star (Rd) in relation to the quotient MWD,f = 0.9M 1100 (¯ρd/ρd,c) for the same system as in Fig. 5. The minimum radius MWD,f = 0.8M Rmin is reached at a mass of 0.20 M in this model sequence. 1000 MWD,f = 0.7M ]

1 900 − 12 800 (B) ˙ 11 Mcrit [km s 700 M˙ 1 ej

] 10 v 1

600 − 9 Md,in = 0.55M yr

500 8 MWD,in = 1.00M

MWD,in = 0.60M M 400 · 7 8

− 6 0.2 0.3 0.4 0.5 0.6 0.7 [10 5 ˙ Md [M ] M 4 a˙/a < 0a ˙/a > 0 3 Fig. 3: Representative model sequences detailing the evolution 2 of individual systems in the Md-vej space. MWD,in is the initial .5 .450 .40 .350 .30 .250 .20 0.150 accretor mass for each of the model sequences with MWD,in = 1.00 M in panel (A) and MWD,in = 0.60 M in panel (B). Each Md [M ] dot represents a single model with terminal accretor mass MWD,f. Velocities given with respect to the center of mass of the progen- Fig. 5: Comparison between the critical mass transfer rate M˙ crit itor binary for increasing a as defined in Eq. 6 and the actual mass transfer rate M˙ 1 for a representative system with respect to donor mass Md. Initial masses Md,in and MWD,in are as indicated.

5.2. The ejection velocity maximum polytropic stellar models with ! As argued previously, the maximum in ejection velocities exhib- ρ¯ 3 dw = − (8) ited by individual model sequences is expected to translate to ρ z dz the full spectra. This indicates that for each assumed value of c z=zn MWD,f there exists a maximum ejection velocity vej,max(MWD,f). Where zn are the solutions to the Lane-Emden equation Consequently, any observed hypervelocity runaway with an in- ! ferred ejection velocity higher than v (M ) for an as- 1 d dw ej,max WD,f z2 + wn = 0 (9) sumed MWD,f must necessarily have been ejected either from z2 dz dz a system with a higher MWD,f or by a different mechanism al- together. Any such observation is especially auspicious in the and n the polytropic index. Low mass WDs are well approxi- case of MWD,f ∼ 1.4 M . The occurrence of this maximum is mated by polytropic equations of state with n = 3/2 (see e.g. driven by the widening of the binary as the donor star becomes Kippenhahn et al. 2012). In the systems under discussion here, more degenerate once its mass drops below the value needed for as the donor star loses mass, the quotientρ ¯d/ρd,c is thus expected sustained helium burning. This is the 0.3 M -limit mentioned to evolve towards values compatible with a polytropic index of above. As the donor star loses sufficient mass to drop below this n = 3/2. An example of this is shown in Fig. 4. It is notewor- thy that, as the donor star approaches ρ¯ /ρ  , its radius limit, the lack of energy generation by nuclear fusion leads to d d,c zn=z3/2 contraction on the thermal timescale concurrent with a reconfig- passes a minimum. This inverse correlation (compared to main uration of the star’s structure to become more degenerate. This sequence stars) of mass and radius is a well known property of reconfiguration is conveniently illustrated by the evolution of the WDs. As the donor star approaches a radius minimum, the mass quotientρ ¯d/ρd,c (i.e. the donor star’s average density divided by transfer rate, as shown in Fig. 5, increases. As shown, this in- its central density). This quotient is a well known quantity in crease in mass transfer is sufficient to exceed the critical mass

Article number, page 7 of 17 A&A proofs: manuscript no. paper

1.016 1 1200

1.15 M < M < 1.25 M 1100 1.3M Md,in = 1.00M , MWD,in = 0.55M 1.014 WD −6 0.999 1000 10 1.012

900 ] ] 1 1 1.2M 1.01 − 0.998 − 800 −7

10 yr d,min ej,max

1.008 700 R 1.1M v / / [km s d [M 0.997 ej 600 RLOF 2 ej R . 1 006 v 1.0M ˙ v 0.9M 500 detaches −8 M 1.004 0.8M 10 0.996 400 0.7M 0.6M 1.002 300 t = t0 1 0.995 200 10−9 0.18 0.19 0.2 0.21 0.22 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Md [M ] Md [M ]

Fig. 6: Mass-radius relationship of donor stars in the vicinity of Fig. 7: Model sequence in Md-v-space of a single binary system the ejection velocity maximum. The mass-dependent donor star with initial masses Md,in and MWD,in, i.e. q > 1. Color indicates radius (Rd) is normalized to the minimum radius of that partic- the current mass transfer rate M˙ . Labels along the graph indicate ular model sequence (Rd,min), the mass dependent ejection ve- the current accretor mass. t = t0 is indicated as well as the points locity, indicated by the color bar, (vej) is analogously normalized where the system detaches and undergoes a second RLOF. Ve- to the maximum ejection velocity of the same model sequence. locity given with respect to the center of mass of the progenitor For clarity, only tracks in the vicinity of an accretor mass in the binary range 1.15 M < MWD < 1.25 M are shown.

mentum loss due to GWR has decreased the orbital separation transfer rate defined in Eq. 6, in turn leading to an increase of sufficiently to initiate a second RLOF. The lower orbital veloci- the orbital separation and, hence, a decrease of the orbital and ties prior to detachment will lead to the presence of a secondary ejection velocity. In the present sample, the radius minimum is branch in the full ejection velocity spectra. However, due to the generally attained in the range 0.19 M < Md < 0.25 M (Fig. 6). limited nature of the initial parameter space in this study, this Objects in this range would, as indicated by Fig 4, be character- secondary branch is only resolved in the spectra corresponding ized as proto-WDs that would, following a period on further con- to the lowest terminal WD masses. traction, form a population of low-mass, high velocity runaway WDs. However, as these objects are likely to only properly settle on the WD cooling sequence a considerable time after ejection 5.4. Complete spectra and, corresponding to their high ejection velocity, a significant Full ejection velocity spectra in the range 1.0 M < MWD,f < distance from their point of origin, observation of such an object 1.5 M are shown in Fig. 8. Note that, as would naively be ex- as a high velocity extremely low mass (ELM) WD is deemed un- pected, the maximum ejection velocity is correlated with MWD,f likely, though not impossible. This scenario is similar to the one and the presence of an ejection velocity maximum in the range proposed by Justham et al. (2009) but has to be tempered with 0.19 M < M < 0.25 M for all values of M . In panel (A) the notion of a large fraction if not all of the currently observed d WD,f with MWD,f = 1.0 M , a bifurcation of the ejection velocity spec- ELM WDs being part of a binary (Brown et al. 2020). trum, as described above, is visible. It should further be noted that, with increasing values of MWD,f the maximum donor star 5.3. Bifurcations mass shown in each panel decreases. This is only partially a con- sequence of the choice of initial parameter space, as systems that The orbital velocity evolution of a system with initial mass ratio do not interact during the helium main sequence of the donor star q > 1 are shown in Fig. 7. As can be seen, the system undergoes are excluded from this plot. Donor stars of high initial mass tend two distinct phases of mass transfer: A "fast" phase of high mass to evolve quickly enough to avoid RLOF during the helium main transfer rates, followed by a "slow" phase of low mass trans- sequence at the initial orbital separations chosen in this study fer rates. In stars only undergoing core burning, the mass trans- and are therefore removed from the sample. For these systems to fer timescale is generally comparable to the donor star’s nuclear produce an SN during the core helium burning stage, the donor timescale (i.e. τnuc ∼ τRLOF), however, in systems with q > 1, an- would have to full its Roche lobe entirely directly subsequent to gular momentum transfer due to RLOF additionally acts to de- the most recent CE phase. crease the system’s orbital separation, leading to enhanced mass Systems in the entire considered range of MWD,f are capa- transfer. However, as the donor star is initially the more massive ble of producing hypervelocity stars, assuming ejection occurs companion in these systems, it will initially orbit with a lower or- in the Solar neighborhood, higher mass runaways (& 0.5 M ) bital velocity. The donor star will lose mass to the accretor until are less likely to become unbound from the Galaxy depending q = 1 is reached, at which point angular momentum transfer will on both the terminal WD mass, the local Galactic escape ve- act to increase the orbital separation (see Eq. 5) until the system locity and ejection direction. As such, the local Galactic escape detaches. Prior to the point of detachment, the orbital velocity velocity should be compared with the ejected companion’s space of the donor star will generally be lower than in systems with velocity immediately following the SN event. Further, as seen in q < 1 at the same donor star mass. Subsequent to detachment, Fig. 8 (E), local Chandrasekhar mass explosions can be expected the components will then evolve in isolation until angular mo- to always produce a hypervelocity hot subdwarf. Ejection veloc-

Article number, page 8 of 17 P. Neunteufel: Hypervelocity star ejection

1300 1300 1200 1200 1100 1100 US 708 (2015) US 708 (2015) 1000 1000 900 900 US 708 (2020) US 708 (2020)

] 800 ] 800 -1 -1 s 700 s 700

[km 600 [km 600 vesc,loc vesc,loc ej ej v 500 v 500 400 400 300 300

200 (A) MWD,t = 1.0 MSun 200 (B) MWD,t = 1.1 MSun 100 100 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

1300 1300 1200 1200 1100 1100 US 708 (2015) US 708 (2015) 1000 1000 900 900 US 708 (2020) US 708 (2020)

] 800 ] 800 -1 -1 s 700 s 700

[km 600 [km 600 vesc,loc vesc,loc ej ej v 500 v 500 400 400 300 300

200 (C) MWD,t = 1.2 MSun 200 (D) MWD,t = 1.3 MSun 100 100 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

1300 1300 1200 1200 1100 1100 US 708 (2015) US 708 (2015) 1000 1000 900 900 US 708 (2020) US 708 (2020)

] 800 ] 800 -1 -1 s 700 s 700

[km 600 [km 600 vesc,loc vesc,loc ej ej v 500 v 500 400 400 300 300

200 (E) MWD,t = 1.4 MSun 200 (F) MWD,t = 1.5 MSun 100 100 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

Fig. 8: Ejection velocity spectra for indicated terminal accretor masses (MWD,t). Md and vej are the terminal mass and expected ejection velocity with respect to the center of mass of the progenitor binary of the remnant runaway hot subdwarf. The blue dots indicate the exact state of the system when the accretor reaches the indicated MWD,t. Lines indicate the evolution of the system in an envelope from MWD,t − 0.025 · M < MWD,t < MWD,t + 0.025 · M . The dotted purple line indicates Mrem = 0.3 M , below which core helium burning ceases. The inferred ejection velocity of US 708 according to Geier et al. (2015) and current proper motions provided by Gaia-DR2 (Gaia Collaboration et al. 2018), represented by the solid red and dashed purple lines respectively, and local Galactic escape velocity according to Piffl et al. (2014), including error bars, are given for orientation. Article number, page 9 of 17 A&A proofs: manuscript no. paper

−1 ities higher than 1000 km s can be expected for terminal WD 1400 masses & 1.1 M . The spread in the ejection velocity spectra (i.e. 1300 the presence of multiple ejection velocities for a single value of Md) is a consequence of the degeneracy of multiple initial sys- 1200 tems for a set of terminal WD masses and terminal donor masses. ]

1 1100 The spread is then a consequence of each donor star in the de- − 1000 US 708 (2015) generate set having lost a different amount of mass to the accre-

[km s 900 tor during a mass transfer episode of different length, leading to US 708 (2020) a slightly different structure and chemical composition in each 800 ej,max

case. However, the spread is small enough that a dependence of v 700 the ejection velocity on the runaway mass is still clearly indi- 600 v cated. The inferred ejection velocity of the runaway hot subd- esc,loc 500 warf US 708 according to Geier et al. (2015) is 998 km s−1. This ejection velocity can be reached by any explosion involving an 400 0.6 0.8 1 1.2 1.4 1.6 1.8 2 accretor mass MWD,f = 1.1 M . However, importantly, only if Md,f > 0.3 M in systems with MWD,f ≥ 1.4 M . MWD,f [M ] With an inferred ejection velocity calculated from the proper v = vmax motions provided in Gaia-DR2, (Gaia Collaboration et al. 2018) v = vmax (M = 0.3 M ) −1 of 897 km s , the terminal accretor mass could be as low as v = vmax (M = 0.4 M ) MWD,f = 0.85 M . In this case, the minimum terminal accretor v = vmax (M = 0.5 M ) mass for Md,f > 0.3 M is, notably, MWD,f = 1.1 M . The case of US 708 will be discussed in greater detail in Sec. 6. Fig. 9: Maxima of all ejection velocity spectra as given in Fig. 8 As seen in Fig. 8 (F), 1.5 M explosions are capable of pro- with respect to terminal accretor mass (MWD,f), where vmax indi- −1 pelling a runaway to velocities up to 1200 km s , which is about cates the absolute maximum and vmax(Md) the maximum, with 100 km s−1 slower than the D6-2 object as found by Shen et al. respect to the center of mass of the progenitor binary, for a given (2018). However, it should be mentioned that the reliability of mass of the ejected companion (Md). Black lines indicate in- this particular measurement is being debated in literature (Scholz ferred ejection velocities as calculated by Geier et al. (2015) and 2018). using proper motions as provided by GAIA DR2 (Gaia Collab- oration et al. 2018). The dashed lines indicate the expected con- tinuation of the maxima towards areas of the parameter space 5.5. Runaway velocities as a probe of the pre-explosion insufficiently covered by the grid. progenitor state

As seen in Sec. 5.4, given a constant value MWD,f, ejection ve- locities are a strong function of Md,f. Fig. 9 shows the maxi- Sec. 5.3, rotational velocities are expected to be uniformly higher −1 −1 mum ejection velocities obtained in this sample with respect to than vf,rot = 270 km s and lower than vf,rot = 326 km s , inde- the terminal WD mass (both the total maximum and the max- pendent of the terminal accretor mass. With respect to the crit- ima for a given minimum mass of the runaway). As the de- ical rotational velocity vrot,crit, this corresponds to a range be- picted values are the theoretical maxima of ejection velocities, tween 0.29 · vrot,crit and 0.33 · vrot,crit. The reason for the unifor- they can be used to obtain constraints on the parameters of the mity of rotational velocities lies in the underlying assumption progenitor binary at the time of explosion. Specifically, since of tidal locking, as donors of equal mass but increasing accretor higher ejection velocities require higher terminal WD masses, masses will find themselves in correspondingly wider systems. but higher terminal donor star masses inhibit them, knowledge This uniformity was also noted by Geier et al. (2015), with the of the ejection velocity provides constraints on the parameter upper limit presented in this study in reasonably good agreement space of both the donor and the accretor mass. US 708 is given with the inferred, model dependent, initial rotational velocity of −1 as an example in this plot, with a terminal WD mass in the vf,rot =∼ 350 km s derived by Geier et al. (2015). The assump- range 1.07 M < MWD,f < 1.39 M with corresponding values tion of tidal locking necessarily leads to a direct correlation of of 0.18 M < MWD,f < 0.31 M for the terminal donor star mass rotational and ejection velocity. The differing rotational velocity based on proper motions obtained by Geier et al. (2015). Based predicted at the lower limit (like the spread seen in the ejection on proper motions obtained from Gaia-DR2 (Gaia Collaboration velocity spectra) indicates that the donor star’s structure and evo- et al. 2018), these values amend to 0.85 M < MWD,f < 1.39 M lutionary history does have a noticeable effect on the final state with corresponding values of 0.18 M < MWD,f < 0.35 M . In of the system. Remarkably, the spread of rotational velocities is both cases assuming sub-Chandrasekhar or Chandrasekhar-mass comparable to that of ejection velocities, as shown in Fig. 8. As explosions. Note that, as calculated by Bauer et al. (2019), the can further be seen, the predicted terminal rotational velocities terminal donor mass may be as much as twice that of the current are larger by at least a factor of 2.3 than the observed current mass of the eventually observed runaway. −1 vrot sini = 115 ± 8km s of US 708. Here it should be borne in mind that the radius of the ejected donor star is unlikely to cor- 5.6. Rotation respond to the rotational velocity post-ejection. This was also noted by Geier et al. (2015). The decrease in velocity can be ex- While the question of the rotational velocity of the ejected HVS plained by the star’s post-ejection evolution on the extreme hori- is not a primary subject in this study, the fact of its poten- zontal branch under conservation of angular momentum without tial accessibility to observation merits a brief discussion. The invocation of SN interaction. However, it should also be empha- terminal surface rotational velocities of the ejected component sized that the post-ejection evolution is likely strongly dependent are shown in Fig. 10. Excepting bifurcations, as discussed in on the question of ongoing nuclear processes inside the star, with

Article number, page 10 of 17 P. Neunteufel: Hypervelocity star ejection stars below the threshold for helium burning reacting differently Table 2: Proper motions utilized in kinematic analysis than stars above this threshold. Therefore, the thermal response to ejecta impact may still be important, as indicated by the results Data set µα cos(δ) µδ presented by Bauer et al. (2019). [mas yr−1] [mas yr−1] 2015 −8.0 ± 1.8 9.1 ± 1.6 2020 −5.363 ± 0.391 1.285 ± 0.382 6. The case of US 708

As mentioned in Sec. 1, the hypervelocity runaway US In the absence of accessible newer data, the values of visual 708 (HV2, SDSS J093320.86+441705.4, Gaia DR2 magnitude (mg = 18.668±0.008mag) and heliocentric radial ve- 815106177700219392) is classified as a helium-rich hot −1 subdwarf. Stars of this type are found at the blue end of the locity (vhelio = 917±7km s ) are adopted unchanged from Geier horizontal branch and are thought, most importantly for the et al. (2015). As the determination of and radial purposes of this study, though not exclusively, to products of velocity does not rely on an adopted mass or luminosity, they close binary evolution. Extensive discussion of the properties of are kept constant in the calculation of the mass-dependent space these objects is beyond the scope of this paper, but for thorough velocity. The reader is cautioned to note that the following kine- reviews, the reader is directed to Heber (2009) and Heber matic analysis is intended to represent an idealized model, ne- (2016). glecting error propagation from observations and the Galactic potential. It is likely that inclusion of errors will detract from the In Geier et al. (2015), a current mass of 0.3 M for US 708, (due to the unavailability of a reliable mass measurement) was unambiguity of the drawn conclusions. adopted, yielding a most likely terminal WD mass of 1.3 M . With the predictions presented in Sec. 5, this assumption can be 6.1. The 2015 data set checked for consistency. This is done via the following prescrip- tion: Following the mass-dependent trajectories back to the Galactic plane-crossing yields an upper limit for the mass of US 708 of 1. The current mass of US 708 is allowed to be a free parame- 0.45 M as any assumed higher mass would result in the trajec- ter. This will, as distance was determined spectroscopically tory avoiding intersection with the plane altogether. A compar- by Geier et al. (2015), impact the determination of the cur- ison of inferred current space and ejection velocities with pre- rent space velocity, now relying solely on proper motion and dicted ejection velocity spectra is shown in Fig. 11. Excluding . additional momentum being imparted on the ejected runaway 2. Using kinematic analysis (See App. B) and assuming ejec- through interaction with the supernova ejecta, an assumed mass tion in the Galactic disc, a mass-dependent inferred ejection of 0.3 M is inconsistent with an terminal WD mass of 1.3 M velocity, taking into account local Galactic rotation, is ob- (Fig. 11 D). Bauer et al. (2019) suggest that ejecta interaction tained. would impart an additional kick of ∼ 180 km s−1 on a runaway 3. The resulting ejection velocities are compared with the ejec- of 0.344 M . Since this kick would be imparted perpendicular tion velocity spectrum for a given terminal WD mass. to the donor’s orbital motion, the ejection velocity would be in- creased by ∼ 17 km s−1, still insufficient to allow for consistency. Regarding the second item, it should be mentioned that assum- It can therefore be concluded that the mass of US 708 would ing ejection in the Galactic disc is not necessarily warranted, as need to be in the range 0.27 M < MUS708 < 0.29 M assuming the star may originate outside the disc, e.g. in a globular cluster, a terminal WD mass of 1.3 M . Assuming a terminal WD mass correspondingly affecting inferred ejection velocities. Assuming of 1.4 M , i.e. close to the Chandrasekhar mass, yields a likely similar structure, mass (M) and luminosity (L) of two stars are range of 0.28 M < MUS708 < 0.3 M , consistent with the mass approximately correlated through the homology relation (see e.g. adopted by Geier et al. (2015). Assuming a super-Chandrasekhar Kippenhahn et al. 2012) mass explosion with a terminal WD mass of 1.5 M would in- dicate a mass range of 0.29 M < MUS708 < 0.32 M , also con- !3 !4 L M µ sistent with the mass adopted by Geier et al. (2015). It can be 1 = 1 1 (10) L M µ concluded that, if US 708 was ejected in a sub-Chandrasekhar 2 2 2 mass SN, then both its terminal and current mass should be with µ the mean atomic weight, which is assumed to remain un- smaller than 0.3 M and it should not currently burn helium. changed in this instance. Using the luminosity as calculated in If, on the other hand, US 708 was ejected in a Chandrasekhar or Eq. 10, the distance can then be adjusted by assuming a constant super-Chandrasekhar mass SN, the question of its current state . depends on the amount of material stripped during the SN event, As observational characterization of US 708 has advanced it being highly likely that its terminal mass was greater than somewhat since 2015, with new proper motion data published 0.3 M . It can further be concluded that the observational prop- in the Gaia data release 2 (Gaia Collaboration et al. 2018), but erties of US 708 in the 2015 data set, including its inferred mass, reliable parallax distances and updated radial velocities still un- is most consistent with an origin in a Chandrasekhar or super- available, the following kinematic analysis is performed using Chandrasekhar mass SN. This is in agreement with the conclu- two distinct sets of proper motion parameters. The first set relies sions of Geier et al. (2015). on the data obtained by ground-based observations published in Geier et al. (2015) (labeled ”2015” throughout the paper), the 6.2. The 2020 data set other on the more recent values obtained by the Gaia instru- ment and published in Gaia Collaboration et al. (2018) (labeled The Gaia-based proper motions imply a significantly lower cur- ”2020” throughout the paper). Numerical values are given in rent space velocity of around 994 km s−1, leading to an inferred Tab. 2. ejection velocity (again assuming ejection in the Galactic disc)

Article number, page 11 of 17 A&A proofs: manuscript no. paper of around 897 km s−1 at a location both closer to Earth and the Fedorova (1990), Neunteufel et al. (2016) and Neunteufel et al. Galactic center than in the 2015 data set (see Fig. B.2). In this (2019). However, Wang & Han (2009), using population syn- case, the maximum terminal donor mass compatible with even- thesis to study essentially the same problem, but limiting them- tual crossing of the disc is found to be 0.575 M . Notably, the selves to Md,f > 0.6 M , seem to find less strongly constrained lower ejection velocity implied by this data set calls into ques- ejection spectra. However, as they do not present ejection veloci- tion, assuming a current mass 0.3 M for US 708, ejection in a ties in relation to terminal accretor mass, it is difficult to pinpoint Chandrasekhar-mass SN, instead pointing to a terminal accre- the reason for this discrepancy. The most likely reason is the in- tor mass of between 1.1 M and 1.2 M (Fig. 11 B and C). clusion of bifurcations in the ejection velocity spectra, leading to This, notably, would imply a SN involving the DDet mechanism a larger spread, as seen in Fig. 8 (A). They further find relatively (see Sec. 2.2) and fall somewhere into the parameter space in- lower ejection velocities for Md,f ∼ 0.6 M and higher than found vestigated by Nomoto (1982a,b), Woosley & Kasen (2011) and possible for Md,f & 1.0 M in this study. As their calculations Neunteufel et al. (2017). Assuming ejection in a Chandrasekhar include the evolution of the primordial main sequence binary, a mass SN, the most likely current mass for US 708 is in the range systematic correlation between Md,init and MWD,init may be intro- 0.34 M < MUS708 < 0.37 M , significantly higher than the as- duced which is absent from the models in this paper. This may sumed mass of 0.3 M . As in the 2015 data set, much rests on explain the discrepancy at lower Md,f. The discrepancy at higher the question of whether US 708 is currently burning helium. If Md,f may be explained by differences in metallicity and rotation, its mass is found to be below 0.3 M , then, discounting errors, both absent from this study. This applicability of the results ob- a Chandrasekhar-mass detonation would be conclusively ruled tained to observed runaway stars has been discussed at length. out. As in the 2015 data set, as seen in Fig. 9, a runaway mass With respect to possible progenitor systems, the following ob- greater than 0.4 M is ruled out for all but significantly super- jects can be commented upon: Vennes et al. (2012) and Geier Chandrasekhar-mass SNe. However, it should be emphasized et al. (2013), independently showed CD − 30◦11223 to contain that the analytical power of correlating ejection velocity spec- a WD with a mass of 0.75 − 0.77 M and a hot subdwarf (sdB) tra and kinematic analysis would be greatly improved if done of 0.44 − 0.48 M in a detached configuration and a period of with distance measurements independent of stellar luminosity P = 0.04897906 ± 0.00000004d. After entering a semidetached and mass estimates, i.e. parallax distances. state, the WD in this system would need to accrete a substantial amount of He from its companion in order to become capable of producing an SN. Interesting in the context of this study is 7. Discussion V445 Pup, a nova-like variable that erupted in late 2000. Ashok Investigations of the runaway velocity of the surviving compan- & Banerjee (2003) argued this object to represent a helium nova ions of WDs undergoing thermonuclear SNe are hampered by event with a WD accretor of 1.35 M Kato et al. (2008) and a the unresolved nature of the most likely explosion mechanism. relatively massive 1.2 − 1.3 M companion (Woudt This is usually accompanied by the physics involved in the pre- et al. 2009). It is currently unknown whether the donor star in this ceding evolution of the WD undergoing mass accretion being system is a giant or a HeMS star. If the donor can be shown to less than certain as well. Less uncertainty is involved in the evo- be a giant, and an SN eventually occurs, then runaway velocities lution of the donor star, which can therefore serve as a conve- would be too low to produce a HVS. If the donor is a HeMS star, nient entry point for the modeling of ejection velocity spectra. then the fact that it is currently undergoing RLOF, indicates that Some of the simplifications employed in this study come with a it would have to have filled its Roche lobe immediately after the number of caveats. While it was shown here that the initial or- end of the most recent CE phase. Further considering the mass of bital separation of the progenitor binary has little effect on the the donor, ejection of a HVS is still unlikely if significant mass expected ejection velocity, previous studies (e.g. Yoon & Langer cannot be ejected from the system prior to the (assumed) detona- 2004b; Kato & Hachisu 2004; Neunteufel et al. 2017) suggest tion of the accretor. Very recently, the discovery of a very short that the idiosyncratic evolution of the mass transfer rate associ- period binary composed of a 0.337±0.015 M helium sdOB and ated with certain initial orbital separations do impact the ignition a 0.545 ± 0.020 M WD was reported by Kupfer et al. (2020). behavior of the accreted material on the WD. Besides calling into This system, if able to produce an SN at all, is unlikely to be question the assumption of conservative mass transfer, as weak able to produce a HVS. helium ignitions may lead to nova-like events, expelling part of Further theoretical exploration of the parameter space should the accumulated helium layer from the system, the ability of the include explicit treatments of the effects of non-conservative accreting WD to accept additional material without triggering a mass transfer and initial metallicity. Variations of the initial or- supernova explosion will, in reality, be limited (see e.g. Nomoto bital separation at low total binary mass should also be consid- 1982a,b; Yoon & Langer 2004b; Neunteufel et al. 2016, 2019). ered. The effects of tidal interaction are likely important as well. The latter condition can realistically be expected to limit the ejection velocity spectra to those terminal accretor masses com- 8. Summary and conclusions patible with the assumed explosion mechanisms. The impact of non-conservative mass transfer, however, is less straightforward. This paper presents a thorough study of the ejection velocity As mass is lost from the system, compared to the conservative spectra for runaway stars resulting from thermonuclear SNe in case, the metal content of the donor star will be higher for any the single degenerate helium donor channel. combination of donor and accretor mass. As metallicity impacts It is seen that the structural behavior of the donor star implies the radius of the donor star (compare Fig. 2 B), the correspond- the existence of a maximum ejection velocity, correlated with the ing orbital, and hence ejection velocity, can be expected to be terminal mass of the donor star. The location, albeit not the value, lower. The results of these calculations could also be impacted of this maximum, is largely independent of the terminal mass of by the effects of tides, especially heating effects, affecting the ra- the accretor, and lies in the range 0.19 M < Md < 0.25 M . The dius of the donor star, again leading to lower ejection velocities value of the maximum ejection velocity, on the other hand, is dic- for the same mass combinations (Applegate & Patterson 1987). tated by the terminal mass of the accreting companion, with val- The results presented here agree well with those of Ergma & ues correlated with higher terminal masses. It is found that maxi-

Article number, page 12 of 17 P. Neunteufel: Hypervelocity star ejection

− mum ejection velocities in excess of 1000 km s 1 can be attained Brooks, J., Schwab, J., Bildsten, L., Quataert, E., & Paxton, B. 2017, ApJ, 843, with terminal accretor masses higher than 1.1 M . This suggests 151 that the SN ejection scenario is able to account for the existence Brown, W. R. 2015, ARA&A, 53, 15 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. J. 2005, ApJ, 622, L33 of objects like the hypervelocity runaway sdO US 708 without a Brown, W. R., Geller, M. J., Kenyon, S. J., Kurtz, M. J., & Bromley, B. C. 2007, need for additional acceleration mechanisms, such as shock in- ApJ, 671, 1708 teraction. Concurrently, the assumed mass, MUS708 = 0.3 M , Brown, W. R., Kilic, M., Kosakowski, A., et al. 2020, ApJ, 889, 49 and inferred ejection velocity of this object in the 2015 data de Jong, R. S., Barden, S. C., Bellido-Tirado, O., et al. 2016, in Proc. SPIE, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, set is most consistent with a Chandrasekhar mass detonation, 99081O while more recently obtained proper motions suggest a sub- de la Fuente Marcos, R. & de la Fuente Marcos, C. 2019, arXiv e-prints, Chandrasekhar detonation with a terminal mass in the range arXiv:1906.05227 Di Stefano, R., Voss, R., & Claeys, J. S. W. 2011, The Astrophysical Journal, 1.1 M to 1.2 M . This result implies that the ejection of US 738, L1 708 is compatible with its progenitor being identified as a sin- Edelmann, H., Napiwotzki, R., Heber, U., Christlieb, N., & Reimers, D. 2005, gle degenerate, helium accreting WD undergoing a SN accord- ApJ, 634, L181 ing to the double detonation mechanism as proposed by Nomoto Eggleton, P. 2011, Evolutionary Processes in Binary and Multiple Stars (Cam- (1982a,b). Assuming a Chandrasekhar mass detonation with the bridge Univ. Press) Eggleton, P. 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Ritter, H. 1988, A&A, 202, 93 Table B.1: Parameters used in Eqs. B.2-B.4. Scholz, R.-D. 2018, Research Notes of the American Astronomical Society, 2, 211 Mb/d/h [MG] ad/h [kpc] bb/d/h [kpc] Λ [kpc] Sesana, A., Madau, P., & Haardt, F. 2009, MNRAS, 392, L31 Bulgeb 409 ± 63 0.23 ± 0.03 Shen, K. J., Boubert, D., Gänsicke, B. T., et al. 2018, ApJ, 865, 15 +376 +0.53 +0.020 Sim, S. A., Röpke, F. K., Hillebrandt, W., et al. 2010, ApJ, 714, L52 Diskd 2856−202 4.22−0.99 0.292−0.025 Soker, N. 2013, in IAU Symposium, Vol. 281, Binary Paths to Type Ia Super- +27933 +25.963 +0.53 +0 Haloh 1018−603 2.562−1.419 4.22−0.99 200−82 novae Explosions, ed. R. Di Stefano, M. Orio, & M. Moe, 72–75 Tauris, T. M. 2015, MNRAS, 448, L6 Tutukov, A. V. & Yungelson, L. R. 1979, Acta Astron., 29, 665 Vennes, S., Kawka, A., O’Toole, S. J., Németh, P., & Burton, D. 2012, ApJ, 759, for the bulge, where R is the distance from the Galactic center L25 Vennes, S., Nemeth, P., Kawka, A., et al. 2017, Science, 357, 680 Md Φd(r,z) = − (B.3) Wang, B. & Han, Z. 2009, A&A, 508, L27 r q 2 2 2 Wang, B., Justham, S., & Han, Z. 2013, A&A, 559, A94 r + (ad + z + bd) Webbink, R. F. 1984, ApJ, 277, 355 Woosley, S. E. & Kasen, D. 2011, ApJ, 734, 38 Woosley, S. E. & Weaver, T. A. 1994, ApJ, 423, 371 for the disk, where r is the distance from the Galactic center in Woudt, P. A., Steeghs, D., Karovska, M., et al. 2009, ApJ, 706, 738 the x-y-plane and z is the distance from the x-y-plane and Yoon, S.-C. & Langer, N. 2004a, A&A, 419, 645 " γ−1 ! γ−1 # Yoon, S.-C. & Langer, N. 2004b, A&A, 419, 623 M 1 1 + (R/a ) (Λ/a ) Φ (R) = − h ln h − h if R < Λ Yoon, S.-C. & Langer, N. 2005, A&A, 435, 967 h γ−1 γ−1 ah γ − 1 1 + (Λ/ah) (1 + Λ/ah) Yu, Q. & Tremaine, S. 2003, The Astrophysical Journal, 599, 1129 − Yungelson, L. R. 2008, Astronomy Letters, 34, 620 M (Λ/a )γ 1 = − h h if R ≥ Λ (B.4) Zhang, M., Fuller, J., Schwab, J., & Foley, R. J. 2019, ApJ, 872, 29 γ−1 R (1 + Λ/ah) with γ = 2 and the other parameters given in Tab. B.1. Appendix A: Effects of initial orbital separation The distance of the Sun from the Galactic center is given as As described in Sec. 4, initial orbital separations a (ξ) in this rSol = 8.40 ± 0.08 kpc. init Using the trajectory solver with the parameters as described paper were chosen such that Eq. 1 satisfies Rd,RL = ξ · Rd with ξ = 1.005. Since this choice results in a variety of initial orbital above, and letting the inferred mass of US 708 be a free pa- separations and initial periods, orbital separations and initial pe- rameter, as described in Sec. 6, the time-of-flight (Fig. B.1) and riods will not be comparable between individual systems, even ejection location (Fig. B.2) can be calculated. between systems with equal total or component masses. As ar- Results for both time-of-flight and ejection location are in gued, the window for RLOF during the HeMS of the donor star good agreement with the more detailed calculations presented by is quite narrow in terms of ξ and the effect of a different choice Geier et al. (2015) for the preferred model of MUS708 = 0.3 M . of that parameter on the expected ejection velocity correspond- ingly small. In order to test this argument, the full grid was rerun with initial orbital separations corresponding to ξ = 1.01. As seen in Fig. A.1, the maximum ejection velocity with ξ = 1.01 is not significantly increased compared with ξ = 1.005 at the same mass with analogous Md,f. It can reasonably concluded that the initial orbital separation is of secondary importance for the question of ejection velocity maxima. However, due to the desire to access the entire mass spectrum with two sets of initial models, this increase is small. While the predictive power hat high values of Md,init can be called reasonable, more work is required in cases of low Md,init.

Appendix B: HVS kinematic analysis Trajectory and time-of-flight data as described in Sec. 6 were calculated by numerically solving the the Newtonian equations of motion in the well known form d dx m = −∇Φ(x) (B.1) dt dt with Φ the Galactic potential. The numerical solver utilized here was newly developed on the basis of a fourth-order Runge-Kutta integrator with adaptive step size control as described by Press et al. (1992). The Galactic potential was assumed to be static and correspond to Model 1 put forward by Irrgang et al. (2013) as a revision of Allen & Santillan (1991) in the form

Mb Φb(R) = − q (B.2) 2 2 R + bb Article number, page 14 of 17 P. Neunteufel: Hypervelocity star ejection

US 708 (inf) US 708 (inf) 350 350 -1 -1 vrot,max = 321 km s vrot,max = 321 km s

300 300

] ] -1 -1 -1 vrot,min = 274 km s s 250 s 250 [km [km -1

rot vrot,min = 230 km s rot v v 200 (A) MWD,t = 1.0 MSun 200 (B) MWD,t = 1.1 MSun

150 150

US 708 (vrot sin i) US 708 (vrot sin i)

100 100 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

US 708 (inf) US 708 (inf) 350 350 -1 -1 vrot,max = 320 km s vrot,max = 319 km s

300 300 ] ]

-1 = 271 km s-1 -1 -1 s 250 vrot,min s 250 vrot,min = 271 km s [km [km rot rot v v 200 (C) MWD,t = 1.2 MSun 200 (D) MWD,t = 1.3 MSun

150 150

US 708 (vrot sin i) US 708 (vrot sin i)

100 100 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

US 708 (inf) US 708 (inf) 350 350 -1 -1 vrot,max = 326 km s vrot,max = 319 km s

300 300 ] ]

-1 -1 -1 -1 s 250 vrot,min = 270 km s s 250 vrot,min = 270 km s [km [km rot rot v v 200 (E) MWD,t = 1.4 MSun 200 (F) MWD,t = 1.5 MSun

150 150

US 708 (vrot sin i) US 708 (vrot sin i)

100 100 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Md [MSun] Md [MSun]

Fig. 10: Same as Fig. 8, but showing the ejected companion’s surface rotational velocity at the time of ejection with the observed current vrot sini, including error bars, given for comparison, wit the red dashed line indicates the inferred surface rotational velocity at ejection according to Geier et al. (2015). Green lines indicate the minimum and maximum velocity found in the sample as labeled.

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1400 1400 1300 vej 1300 vej vtr,2015 vtr,2015 vtr,2020 vtr,2020 1200 vgrf,2015 1200 vgrf,2015 vgrf,2020 Geier+ 2015 vgrf,2020 Geier+ 2015 1100 1100 s] s] / / 1000 1000 [km [km v 900 v 900 800 800

700 (A) MWD(t = tSN) = 1.0 M 700 (B) MWD(t = tSN) = 1.1 M 600 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

M [M ] M [M ] 1400 1400 1300 vej 1300 vej vtr,2015 vtr,2015 vtr,2020 vtr,2020 1200 vgrf,2015 1200 vgrf,2015 vgrf,2020 Geier+ 2015 vgrf,2020 Geier+ 2015 1100 1100 s] s] / / 1000 1000 [km [km v 900 v 900 800 800

700 (C) MWD(t = tSN) = 1.2 M 700 (D) MWD(t = tSN) = 1.3 M 600 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

M [M ] M [M ] 1400 1400 1300 vej 1300 vej vtr,2015 vtr,2015 vtr,2020 vtr,2020 1200 vgrf,2015 1200 vgrf,2015 vgrf,2020 Geier+ 2015 vgrf,2020 Geier+ 2015 1100 1100 s] s] / / 1000 1000 [km [km v 900 v 900 800 800

700 (E) MWD(t = tSN) = 1.4 M 700 (F) MWD(t = tSN) = 1.5 M 600 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

M [M ] M [M ]

Fig. 11: Inferred current Galactic rest frame velocity, vgrf(t = t0), and ejection velocity, vtr(t = tSN), of US 708, as dictated by observed radial velocity and proper motion, correlated with the ejection velocity spectra, vej(t = tSN), with a WD explosion mass (MWD(t = tSN)) as labeled. Observed space and ejection velocities are base either on Geier et al. (2015) (denoted by subscript ”2015”) or on proper motions according to Gaia DR2 (subscript ”2020”, Gaia Collaboration et al. 2018) Spectra are represented by the envelopes of their depictions in Fig. 8 for legibility. The current space and inferred ejection velocities, as determined by Geier et al. (2015) are depicted as purple dots.

Article number, page 16 of 17 P. Neunteufel: Hypervelocity star ejection

1400 30 0.45 1300 0.4 1200 20 ]

1 1100 0.35

− (A) PM(2015) US 708 (2015) 10 ]

1000 0.3 [M

[km s 900 US 708 (2020) 0 Gal.Cen. 800 [kpc] M

y US708 0.25 US708 ej,max

v 700

= M −10 0 .3 600 v M 0.2 esc,loc 500 − 20 0.15 400 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −30 0.1 MWD,f [M ] −30 −20 −10 0 10 20 30 v = v max x [kpc] v = vmax (M = 0.3 M ) v = vmax (M = 0.4 M ) 30 0.55 v = vmax (M = 0.5 M ) 0.5 20 Fig. A.1: Like Fig. 9, but ainit(ξ = 1.01) (See Sec 4). The gray 0.45 line indicates vmax as in Fig. 9 for comparison. (B) PM(2020)

10 0.4 ]

0.35 [M 50 0 Gal.Cen.

PM 2015 [kpc] 45 M 0.3 y US708

PM 2020 US708 40

= M −10 0 0.25 .3 35 M 30 0.2 −20 25 0.15 20 −30 TOF [Myr] 0.1 15 −30 −20 −10 0 10 20 30 10 x [kpc] 5 MUS708 = 0.3 M 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Fig. B.2: Current inferred location, depending on assumed mass (MUS708) of US 708, and inferred origins relative to the Galac- MUS708 [M ] tic center and the position of the Sun in the Galactic x-y plane. Panel (A) uses proper motions as obtained by Geier et al. (2015), Fig. B.1: Mass-dependent time of flight (TOF) since ejection panel (B) uses proper motions obtained by GAIA (Gaia Collab- from the disc for US 708, once for proper motions as obtained oration et al. 2018) The location, origin and past trajectory of the by Geier et al. (2015) and once for proper motions as published preferred model with MUS708 = 0.3 M is highlighted in green. by GAIA DR2 (Gaia Collaboration et al. 2018). The preferred model with MUS708 = 0.3 M is highlighted in green. Note that the counter-intuitively shorter TOF at similar mass obtained for the slower space velocity of the 2020 data set is a result of the shorter distance traveled to the disc-crossing point. This shorter distance, in turn, is derived from the reoriented direction of travel as compared to the 2015 data set (See Fig. B.2).

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